62,492 research outputs found
Approximation of Eigenfunctions in Kernel-based Spaces
Kernel-based methods in Numerical Analysis have the advantage of yielding
optimal recovery processes in the "native" Hilbert space \calh in which they
are reproducing. Continuous kernels on compact domains have an expansion into
eigenfunctions that are both -orthonormal and orthogonal in \calh
(Mercer expansion). This paper examines the corresponding eigenspaces and
proves that they have optimality properties among all other subspaces of
\calh. These results have strong connections to -widths in Approximation
Theory, and they establish that errors of optimal approximations are closely
related to the decay of the eigenvalues.
Though the eigenspaces and eigenvalues are not readily available, they can be
well approximated using the standard -dimensional subspaces spanned by
translates of the kernel with respect to nodes or centers. We give error
bounds for the numerical approximation of the eigensystem via such subspaces. A
series of examples shows that our numerical technique via a greedy point
selection strategy allows to calculate the eigensystems with good accuracy
Formal Representation of the SS-DB Benchmark and Experimental Evaluation in EXTASCID
Evaluating the performance of scientific data processing systems is a
difficult task considering the plethora of application-specific solutions
available in this landscape and the lack of a generally-accepted benchmark. The
dual structure of scientific data coupled with the complex nature of processing
complicate the evaluation procedure further. SS-DB is the first attempt to
define a general benchmark for complex scientific processing over raw and
derived data. It fails to draw sufficient attention though because of the
ambiguous plain language specification and the extraordinary SciDB results. In
this paper, we remedy the shortcomings of the original SS-DB specification by
providing a formal representation in terms of ArrayQL algebra operators and
ArrayQL/SciQL constructs. These are the first formal representations of the
SS-DB benchmark. Starting from the formal representation, we give a reference
implementation and present benchmark results in EXTASCID, a novel system for
scientific data processing. EXTASCID is complete in providing native support
both for array and relational data and extensible in executing any user code
inside the system by the means of a configurable metaoperator. These features
result in an order of magnitude improvement over SciDB at data loading,
extracting derived data, and operations over derived data.Comment: 32 pages, 3 figure
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